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Transcript
DG4GSP_897_04.qxd 12/20/06 1:08 PM Page 61 Lesson 4.5 • Are There Other Congruence Shortcuts? In this investigation you will investigate two other potential congruence shortcuts. Investigation 1: Is ASA a Congruence Shortcut? If two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle (ASA), must the two triangles be congruent? Sketch Step 1 Open the sketch Congruence Shortcuts.gsp to the page ASA. C C C C A B A B Givens: A A B B Step 2 Drag the points labeled C in the broken triangle on the left so that they coincide to form a triangle. Step 3 In the broken triangle on the right, try to make the points labeled C coincide so that the triangle formed is not congruent to the triangle on the left. Step 4 Change the measure of one or more of the given angles or the given side (the angles and segment below the triangles) and try the experiment again. Investigate 1. Can you form two triangles with different sizes or shapes given the two angles and the side between them? 2. If you are given two triangles such that two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle (ASA), is that enough information to determine that the triangles are congruent? 3. Write a conjecture that summarizes your findings (ASA Congruence Conjecture). (continued) Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press CHAPTER 4 61 DG4GSP_897_04.qxd 12/20/06 1:08 PM Page 62 Lesson 4.5 • Are There Other Congruence Shortcuts? (continued) Investigation 2: Is AAA a Congruence Shortcut? If the three angles of one triangle are congruent to the three angles in another triangle (AAA), must the two triangles be congruent? Sketch Step 1 Go to the page AAA. C A (drag) B (drag) Givens: A B C (Angles A and B determine angle C.) Step 2 Drag the points labeled A and B in the triangle. Step 3 Change the measure of one or more of the given angles (the free angles below the triangles) and try the experiment again. Investigate 1. Can you form two or more triangles with different sizes or shapes given the three angles? 2. If you are given two triangles such that the three angles in one triangle are congruent to the three angles in the other triangle (AAA), is that enough information to determine that the triangles are congruent? 3. Summarize your findings. 62 CHAPTER 4 Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press